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In other words, if C is the centroid of the base, the distance from C to a vertex of the base is twice that from C to the midpoint of an edge of the base. This follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other (see proof).
The vertices of triangles are associated not only with spatial position but also with other values used to render the object correctly. Most attributes of a vertex represent vectors in the space to be rendered. These vectors are typically 1 (x), 2 (x, y), or 3 (x, y, z) dimensional and can include a fourth homogeneous coordinate (w).
A point in the interior of a triangle is the center of an inellipse of the triangle if and only if the point lies in the interior of the triangle whose vertices lie at the midpoints of the original triangle's sides. [3]: p.139 For a given point inside that medial triangle, the inellipse with its center at that point is unique. [3]: p.142
The following two formulas ... the x and y coordinates of the vertex centroid are the ... The four smaller triangles formed by the diagonals and sides of a convex ...
Trilinear coordinates are an example of homogeneous coordinates. The ratio x : y is the ratio of the perpendicular distances from the point to the sides (extended if necessary) opposite vertices A and B respectively; the ratio y : z is the ratio of the perpendicular distances from the point to the sidelines opposite vertices B and C ...
One way to write down a regular n-simplex in R n is to choose two points to be the first two vertices, choose a third point to make an equilateral triangle, choose a fourth point to make a regular tetrahedron, and so on. Each step requires satisfying equations that ensure that each newly chosen vertex, together with the previously chosen ...
Homogeneous coordinates are not uniquely determined by a point, so a function defined on the coordinates, say (,,), does not determine a function defined on points as with Cartesian coordinates. But a condition f ( x , y , z ) = 0 {\displaystyle f(x,y,z)=0} defined on the coordinates, as might be used to describe a curve, determines a condition ...
[au] The tetrahedrally diminished dodecahedron has 16 vertices and 16 faces: 12 trapezoid faces and four equilateral triangle faces. Since the vertex figure of the 120-cell is the tetrahedron, [bl] each truncated vertex is replaced by a tetrahedron, leaving 120 tetrahedrally diminished dodecahedron cells and 120 regular tetrahedron cells. The ...